Seismic data compression by an adaptive local cosine/sine transform and its effects on migration

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1 Geophysical Prospecting, 2000, 48, 1009±1031 Seismic data compression by an adaptive local cosine/sine transform and its effects on migration Yongzhong Wang 1 and Ru-Shan Wu 1 Abstract The local cosine/sine basis is a localized version of the cosine/sine basis with a window function which can have arbitrary smoothness. It has orthogonality and good time and frequency localization properties. The adaptive local cosine/sine basis is a best-basis obtained from an overabundant library of cosine/sine packets based on a costfunctional. We propose a 2D semi-adaptive (time-adaptive or space-adaptive) local cosine transform (referred to as a 2D semi-alct) and apply it to the SEG±EAEG salt model synthetic data set for compression. From the numerical results, we see that most of the important features of the data set can be well preserved even in the high compression ratio (CRˆ40:1) case. Using reconstructed data from the highly compressed ALCT coefficients (CRˆ40:1) for migration, we can still obtain a highquality image including subsalt structures. Furthermore, we find that the window partition, generated by the 2D semi-alct, is well adapted to the characteristics of the seismic data set, and the compression capability of the 2D semi-alct is greater than that of the 2D uniform local cosine transform (2D ULCT). We find also that a (32, 32) or (32, 64) minimum (time, space) window size can generate the best compression results for the SEG±EAEG salt data set. Introduction In seismic data compression, the problem is the selection of suitable bases to represent seismic signals efficiently and therefore to achieve maximum possible compression, while still preserving useful information. This problem has attracted considerable attention (e.g. Donoho, Ergas and Villasenor 1995; Bradley, Fei and Hildebrand 1996; Villasenor, Ergas and Donoho 1996; Wang and Pann 1996; Vassiliou and Wickerhauser 1997; Wu and Wang 1999) and much progress has been made in the past few years. For example, Vassiliou and Wickerhauser (1997) used bi-orthogonal wavelets as the expansion bases for the compression of data sets from Egypt and Trinidad. Their numerical results show that long filters, moderate decomposition depths and frequency-weighted, variance-adjusted quantization yield the best results. Received April 1999, revision accepted March Institute of Tectonics, University of California, Santa Cruz, CA 95064, USA. q 2000 European Association of Geoscientists & Engineers 1009

2 1010 Y. Wang and R.-S. Wu Figure 1. A schematic representation of 2D time-alct, showing uniform LCT along the space axis, and within each fixed space strip, adaptive LCT along the time axis. From their conclusions, we see that compression effectiveness varies considerably for different expansion bases. Local cosine/sine bases are orthogonal bases with good localization in both the space/time and the wavenumber/frequency domains. The adaptive local cosine/sine basis is a best-basis obtained from an overabundant library of cosine/sine packets based on a cost-functional. The adaptability results in a flexible space and/or time segmentation to match the signal characteristics. By contrast, the uniform local cosine transform (ULCT) cannot generate an adaptive matching to a given signal because the width of each window is fixed and equal. We propose a 2D semi-adaptive local cosine transform (either adaptive in the time direction and uniform in the space direction, referred to as 2D time-alct, or adaptive in the space direction and uniform in the time direction, referred to as 2D space-alct) and apply it to seismic data compression of the SEG±EAEG salt model synthetic data set Figure 2. A schematic representation of 2D space-alct, showing uniform LCT along the time axis, and within each fixed time strip, adaptive LCT along the space axis.

3 Adaptive local cosine/sine transform 1011 Figure 3. A sketch drawing of folding on the overlapping zone. b I (x) is a bell function whose nominal support is [a,b]. 1 is the left overlapping radius and 1 0 is the right overlapping radius. Figure 4. A 1D adaptive local cosine multilevel decomposition tree for the 200th trace of the SEG±EAEG salt data set. The preset maximum decomposition level is 4 and Shannon entropy is chosen as the cost-functional.

4 1012 Y. Wang and R.-S. Wu Figure 5. Adaptive window segmentation of the 200th trace of the SEG±EAEG salt data set. The minimum window width is 64 samples, the smoothness parameter n in (9) is 2, the preset maximum decomposition level is 4 and Shannon entropy is chosen as the cost-functional. The dotted lines indicate the edges of the nominal supports of the windows. The solid line shows the input trace. as a test example. The 1D ALCT can be implemented with a fast algorithm of computational complexity O(Nd log 2 N) (Coifman and Wickerhauser 1992), where N is the number of samples in the signal and d is the maximum decomposition level. Adaptive local cosine/sine transform Brief description of the local cosine/sine basis Local cosine/sine bases constructed by Coifman and Meyer (1991) (see also Pascal, Guido and Wickerhauser 1992) consist of cosines/sines multiplied by smooth, compactly supported bell functions. These localized cosine/sine functions remain orthogonal and have small Heisenberg products. The local cosine/sine transform has much in common with the windowed or short-time Fourier transform (WFT or

5 Adaptive local cosine/sine transform 1013 STFT) (Gabor 1946). However, for the latter, the Balian±Low obstruction (Balian 1981) prevents the windowed exponential basis from simultaneously being a frame and having a finite Heisenberg product. Local cosine or sine wavelets, on the other hand, overcome this limitation. The basis element (here, for example, the basis of the fourth type) can be characterized by position a, interval I and frequency/wavenumber index k as follows: or r 2 c aik x ˆ b I x cos p k 1 x 2 a jij 2 jij r 2 c aik x ˆ b I x sin p k 1 x 2 a ; 2 jij 2 jij where b ˆ b I x is a bell function which is a smooth function compactly supported in the interval a 2 1; b 1 0 Š for a 1 # b : This interval contains I ˆ a; bš which is called its nominal support. We can control the window width by I, the left endpoint of the nominal support by the position a and the frequency/wavenumber by the index k. The properties of the bell function, i.e. 8 b I x 2 b I 2a 2 x 2 ˆ 1; x [ a 2 1; a 1Š; >< b I x 2 b I 2b 2 x 2 ˆ 1; x [ b ; b 1 0 Š; 3 >: b I x ˆ1; x [ a 1; b ; ensure the orthonormality of the bases and afford a fast algorithm. Suppose a sequence {a j } is selected to satisfy a j, a j 1 ; lim j!^1 {a j } ˆ ^1 and there also exists an accompanying sequence {1 j } such that a j 1 j # a j j 1 for all j [ Z: Then the functions s 2 c jk x ˆ b aj ;a a j 1 2 a j 1 Š x cos p k 1 x 2 aj ; 4 j 2 a j 1 2 a j j [ Z; k ˆ 0; 1; 2; ¼ ; with positive polarity at a j and negative polarity at a j+1,or s 2 c jk x ˆ b aj ;a a j 1 2 a j 1 Š x sin p k 1 x 2 aj ; 5 j 2 a j 1 2 a j 1 j [ Z; k ˆ 0; 1; 2; ¼ ; with negative polarity at a j and positive polarity at a j+1,form an orthonormal basis for the space L 2 (R). A two-dimensional local cosine/sine basis can be generated by the tensor products

6 1014 Y. Wang and R.-S. Wu Figure 6. Window partition, generated by the 2D time-alct, for the SEG±EAEG data set. The minimum time-window width is 32 samples and the uniform space-window width is 64 traces. Shannon entropy is selected as the cost-functional. c ax I x k x x c ay I y k y y and their nominal supports are the Cartesian product rectangles of the nominal supports of the x and y factors. Adaptive local cosine/sine basis 1D ALCT For binary decomposition, if we fix the window width I jij ˆja j 1 2 a j j ; all wavelets in (4) or (5) can form an orthonormal basis for L 2 (R) and this basis is called level 1; if we then select I /2 as the window width, the wavelets in (4) or (5) can also form an orthonormal basis and this is called level 2; and so on. Thus, a binary-based signal decomposition tree consists of the bases at different levels. However, not all the bases are efficient in matching a given signal. Therefore, we must pick the `best-basis' from all the possible local cosine/sine bases, using a cost-functional. To search for the local cosine/sine best-basis, i.e. an adaptive local cosine/sine basis, in order to achieve the best matching to the signal, a cost-functional is defined based

7 Adaptive local cosine/sine transform 1015 Figure 7. Window partition, generated by the 2D space-alct, for the SEG±EAEG data set. The uniform time-window width is 32 samples and the minimum space-window width is 32 traces. Shannon entropy is selected as the cost-functional. on the entropy of the decomposed signal. There are numerous cost-functionals cited in the literature (see for example Misiti et al. 1996). Here, we use Shannon entropy as the cost-functional, i.e. E X ˆ2 X p n log 2 p n ; 6 n where p n ˆjx n j 2 =kxk 2 ; X is the signal of length N samples, x n is the nth component of X and E(X) denotes the entropy of the signal X. In implementation, we use Coifman and Wickerhauser's (1992) fast algorithm to search for the best-basis based on the Shannon entropy. The main concept of this algorithm is that the full local cosine/sine tree is pruned recursively at each node by comparing its entropy to the summation of the entropy of its corresponding child nodes, i.e. IF Entropy parent node # Entropy child1 Entropy child2 Š THEN cut off the child branches: Initially, a full binary-based decomposition tree with a preset maximum

8 1016 Y. Wang and R.-S. Wu decomposition level is produced. The pruning procedure then starts from the leaf nodes and proceeds towards the root. At the end of this procedure, an optimal pruned tree is obtained for the given signal, i.e. an adaptive local cosine/sine basis is obtained. For more details, see Coifman and Wickerhauser (1992). 2D semi-alct For the 2D case, we propose two semi-adaptive schemes, i.e. 2D time-alct and 2D space-alct. The 2D time-alct is uniform in the space direction and adaptive in the time direction. Its adaptability is accomplished by the above-mentioned 1D ALCT algorithm along the time direction for a strip with fixed width (see Fig. 1). For the 2D space-alct, the segmentation is uniform in the time direction but adaptive in the space direction (see Fig. 2). Implementation Figure 8. SEG±EAEG salt velocity model A±A' (velocity in feet/second). Bell function Generally, the bell function over I ˆ a; bš is defined by b I x ˆS 1 x 2 a C 1 0 x 2 b ; x [ R; 7

9 Adaptive local cosine/sine transform 1017 Figure 9. The zero-offset seismic section from the model generated by exploding reflector simulation. where S 1 x 2 a ; sin u n and x2b C1 0 x 2 b ; cos u n : We select 8 >< u 0 x ˆ and >: 8 >< u n x ˆ >: x2a 1 0; if x,21; p 1 x ; 4 if 2 1 # x # 1; u n21 p ; if x. 1; 2 0; if x,21; sin p 2 x ; if 2 1 # x # 1; p ; if x. 1: Using induction we can show that S 1 x 2 a has 2 n 2 1 vanishing derivatives at x ˆ a 2 1 and x ˆ a 1; and C 1 0 x 2 b has 2 n 2 1 vanishing derivatives at x ˆ

10 1018 Y. Wang and R.-S. Wu Figure 10. Reconstructed data with compression ratio (CR) of 26:1. b and x ˆ b 1 0 : So S 1 x 2 a or C 1 0 x 2 b can be made into an arbitrarily smooth cut-off function. We take n ˆ 2: Folding Rather than calculating inner products with the sequences c a Ik, we can preprocess data so that the standard fast discrete cosine transform of the fourth type (DCT-IV) or discrete sine transform of the fourth type (DST-IV) algorithm may be used. This can be realized by folding the overlapping parts of the bell functions back into the intervals. Suppose we wish to fold a signal f(x) back into the interval I ˆ a; bš across a and b (see Fig. 3). Using the bell function b I (x) defined by (7), we have (using the local cosine transform of the fourth type (LCT-IV) as an example) 8 f x ˆb I x f x b I 2a 2 x f 2a 2 x ; if a # x # a 1; >< f new x ˆ f 2 x ˆb I x f x 2 b I 2b 2 x f 2b 2 x ; if b # x # b; >: f x ; if a 1, x, b : 10 The resultant folded data f new (x) is now defined in the interval [a,b].

11 Adaptive local cosine/sine transform 1019 Figure 11. Residual data between Fig. 9 and Fig. 10 for CR ˆ 26:1 (amplified by a factor of 10). To reconstruct f(x) from f new (x), we can use the following unfolding formulae (again using the LCT-IV as an example): 8 b I x f new x 2 b I 2a 2 x f new 2a 2 x ; if a # x # a 1; >< f x ˆ b I x f new x b I 2b 2 x f new 2b 2 x ; if b # x # b; >: f new x ; if a 1, x, b : 11 Edge extension When the bell shifts to the leftmost endpoint or the rightmost endpoint of the signal, we cannot directly obtain f + (x)orf 2 (x) from the above formulae because of the lack of data in the leftmost and rightmost overlapping zones. Usually, we have four extension schemes: (1) zero-extension; (2) symmetry-extension; (3) smoothness-extension; (4) periodization-extension. In this work, based on the features of seismic signals, we chose the zero-extension.

12 1020 Y. Wang and R.-S. Wu Figure 12. Reconstructed data with CR ˆ 40:1. DCT-IV/DST-IV After the procedures of folding and edge extension, we can apply the fast DCT-IV/ DST-IV to the folded data f new (x) to obtain the local cosine/sine transform coefficients. Searching for the best-basis After we have completed local cosine/sine transforms for all the preset decomposition levels and also calculated the entropy of each transformed subsignal, we can use the Coifman and Wickerhauser (1992) fast algorithm to search for the local cosine/sine best-basis. As an example, the 1D ALCT is applied to the 200th trace of the SEG±EAEG salt data set (the original trace length was 626, but is extended here to 1024 by zero-padding). The minimum window width for this trace is preset at 64 samples, the smoothness parameter n in (9) is chosen as 2, and Shannon entropy is adopted as the cost-functional. Figure 4 indicates its adaptive local cosine multilevel decomposition tree (the preset maximum level is 4). Figure 5 shows its adaptive window segmentation of the trace resulting from Fig. 4. In Fig. 5, the dotted lines

13 Adaptive local cosine/sine transform 1021 Figure 13. Residual data between Fig. 9 and Fig. 12 for CR ˆ 40:1 (amplified by a factor of 10). indicate the edges of the nominal supports of the windows while the solid line shows the input trace. Figures 6 and 7 show the window partitions generated by the 2D semi-alct for the SEG±EAEG data set (in Fig. 6 by the 2D time-alct and in Fig. 7 by the 2D space-alct). In Fig. 6, the minimum time-window width is 32 samples and the uniform space-window width is 64 traces, while in Fig. 7, the uniform time-window width is 32 samples and the minimum space-window width is 32 traces. In Figs 6 and 7, the adaptive grids show the nominal supports of the windows of a 2D best local cosine basis. As can be seen, the adaptive grids match the seismic data well, i.e. along the time or space direction the window widths are shorter where the density of seismic signals is greater. Data compression and imaging using compressed data We now test the performances of our compression schemes (2D time-alct and 2D space-alct) and compare the results with that of 2D ULCT. We also test

14 1022 Y. Wang and R.-S. Wu Figure 14. The coefficient profile with CR ˆ 40:1 after the 2D time-alct of input data in Fig. 9. In this figure, the horizontal axis represents position-wavenumber while the vertical axis represents time-frequency. The minimum (time, space) window size for the 2D time-alct is (16, 16) and Shannon entropy is adopted. the effects of seismic data compression on the image quality obtained by depth migration. Figure 8 shows a 2D profile A±A' of the SEG±EAEG salt velocity model and Fig. 9 shows the synthetic zero-offset data from the salt model using a finite-difference exploding reflector modelling algorithm, generated at AMOCO. This is a very complicated model with a large velocity contrast (salt velocity is as high as three times that of the surrounding media). Figure 10 shows the data reconstructed from the compressed 2D time-alct coefficients with compression ratio CR ˆ 26:1. Compression is achieved by setting a threshold of 1% of the maximum absolute ALCT coefficient for each space-window. Therefore the threshold varies for different spacewindows. Figure 11 shows the corresponding residual data amplified 10 times to show detail. Reconstructed data for the CR ˆ 40:1 case (also using the 2D time-alct) is shown in Fig. 12 and its residual data (amplified by a factor of 10) is plotted in Fig. 13. Here, a threshold of 2% of the maximum absolute ALCT coefficient for each spacewindow is adopted. Figures 10±13 show that the majority of the important features of

15 Adaptive local cosine/sine transform 1023 Figure 15. Hybrid pseudo-screen migration on the original synthetic data. the original synthetic data have been preserved very well even in the case of a high compression ratio (CR ˆ 40:1). In implementing the 2D time-alct with the SEG±EAEG salt data set, we select the Shannon entropy as the cost-functional, the maximum decomposition level in the time direction is 6, the uniform window width in the space direction is 16 traces, i.e. the size of the minimum (time, space) window is (16, 16) (the original seismic data is extended to 1024 in time and 2048 in space by zero-padding), the overlapping radius in the time or space direction is one half of the minimum or uniform window width, i.e. (8, 8), and the fixed folding style is used. Also an automatic gain control (AGC) is applied to the data to preserve low-amplitude seismic signals before applying the 2D time-alct. The AGC is subsequently removed after reconstruction. Figure 14 shows the coefficient profile with CR ˆ 40:1 after the 2D time-alct of input data in Fig. 9 using the (16, 16) minimum (time, space) window. Figures 15±19 show the results of migrating the original data and reconstructed data after compression in order to investigate the influence of data compression on the migrated image. Here, the imaging method used is the hybrid pseudo-screen

16 1024 Y. Wang and R.-S. Wu Figure 16. Hybrid pseudo-screen migration on the reconstructed synthetic data from the CR ˆ 26:1 ALCT coefficients. migration (Jin, Wu and Peng 1998). Figure 15 shows migration on the original synthetic data, and Fig. 16 shows migration on the reconstructed data from the compressed coefficient data with CR ˆ 26:1. Figure 18 is the image obtained from migration on the compressed data with CR ˆ 40:1. Figures 17 and 19 show the corresponding residual images after amplification by a factor of 10. Figures 15±19 show that high-quality images including subsalt structures can still be obtained even in the high compression ratio (CR ˆ 40:1) case. To compare the compression capability of the 2D semi-alct with that of the 2D ULCT, we define the signal-to-noise ratio (SNR) as follows: X SNR ˆ 10 log 10 jc k j 2 = X j1 k j!; 2 12 k k where c k is the coefficient above the threshold in the absolute value sense and is thus retained, and 1 k is the coefficient which is discarded. Figure 20 shows the comparison of compression performance between the 2D time-alct and the 2D ULCT. From Fig. 20, we see that the compression capability of the 2D time-alct is more powerful than that of the 2D ULCT.

17 Adaptive local cosine/sine transform 1025 Figure 17. Residual image between Fig. 15 and Fig. 16 for CR ˆ 26:1 (amplified by a factor of 10). To address the relationship between the SNR (in db) of the transformed data and the compression ratio with different minimum (time, space) windows (here, adaptive along the time axis), we tested six different minimum (time, space) windows: (8, 8), (16, 8), (16, 16), (32, 16), (32, 32) and (32, 64). From the curves of SNR versus compression ratio shown in Fig. 21, we see that either (32, 32) or (32, 64) is the best choice of (time, space) windows for the SEG±EAEG data set. Finally, the compression performances of the 2D time-alct and the 2D space- ALCT were compared. In implementing the 2D space-alct, we selected only the (32, 32) and (32, 64) minimum (time, space) windows as tests because these two windows are already the best choices in the 2D time-alct for the SEG±EAEG salt data set. Figure 22 shows that the compression capabilities of the 2D time-alct and the 2D space-alct are almost the same. Conclusions We proposed a 2D semi-adaptive local cosine/sine transform to achieve data compression and we applied it to the SEG±EAEG salt data set. From the numerical

18 1026 Y. Wang and R.-S. Wu Figure 18. Hybrid pseudo-screen migration on the reconstructed synthetic data from the CR ˆ 40:1 ALCT coefficients. results, it was found that the majority of the important features of the original data set can be preserved very well even in the high compression ratio (CR ˆ 40:1) case. Furthermore, even using the highly compressed data (CR ˆ 40:1) for migration, we can still obtain high-quality seismic images including the subsalt structures. We found that the window partitions, generated by the 2D semi-alct, aligned well with natural breaks in the seismic data, therefore the compression capability of the 2D semi-alct is greater than that of the 2D ULCT. Also, the 2D time-alct and 2D space-alct are equally powerful for data compression. Moreover, we found that a (32, 32) or (32, 64) minimum (time, space) window size generated the best compression result for the SEG±EAEG salt data set. Acknowledgements This work was supported by the WTOPI (Wavelet Transform On Propagation and Imaging for seismic exploration) project at the University of California, Santa Cruz. We are grateful to the sponsors. The help from Dr Shengwen Jin on migration is

19 Adaptive local cosine/sine transform 1027 Figure 19. Residual image between Fig. 15 and Fig. 18 for CR ˆ 40:1 (amplified by a factor of 10). greatly appreciated. The facility support from the W. M. Keck Foundation is also acknowledged. Contribution number 402 of the Institute of Tectonics, University of California, Santa Cruz. References Balian R Un principle d'incertitude en theâorie du signal ou en meâcanique quantique. Comptes Rendus de l'academie des Sciences, Paris, SeÂrie II 292, 1357±1362. Bradley J., Fei T. and Hildebrand S Wavelet compression for 3D depth migration. 66th SEG meeting, Denver, USA, Expanded Abstracts, 1627±1629. Coifman R.R. and Meyer Y Remarques sur l'analyse de Fourier aá feneãtre. Comptes Rendus de l'academie des Sciences, Paris, SeÂrie I 312, 259±261. Coifman R.R. and Wickerhauser M.V Entropy-based algorithms for best basis selection. IEEE Transactions on Information Theory 38, 713±718. Donoho P.L., Ergas R.A. and Villasenor J.D High-performance seismic trace compression. 65th SEG meeting, Expanded Abstracts, Houston, USA, 160±163.

20 1028 Y. Wang and R.-S. Wu Figure 20. Compression performance comparison between the 2D time-alct and the 2D ULCT: minimum (time, space) window (16, 16) for 2D time-alct, uniform (time, space) window (16, 16) for 2D ULCT. Shannon entropy is used as the cost-functional for both schemes.

21 Adaptive local cosine/sine transform 1029 Figure 21. Compression performance comparison of different minimum (time, space) windows by 2D time-alct using Shannon entropy as the cost-functional.

22 1030 Y. Wang and R.-S. Wu Figure 22. Compression performance comparison between the 2D time-alct and the 2D space-alct.

23 Adaptive local cosine/sine transform 1031 Gabor D Theory of communication. Journal of the Institute of Electrical Engineers 93(III), 429±457. Jin S., Wu R.S. and Peng C Prestack depth migration using a hybrid pseudo-screen propagator. 68th SEG meetng, New Orleans, USA, Expanded Abstracts, 1819±1822. Misiti M., Misiti Y., Oppenheim G. and Poggi J.M Wavelet Toolbox for Use with MATLAB, pp. 5±7. The Mathworks, Inc. Pascal A., Guido W. and Wickerhauser M.V Local sine and cosine bases of Coifman and Meyer and the construction of smooth wavelets. In: Wavelets: A Tutorial in Theory and Applications (ed. Charles K. Chui), pp. 237±256. Academic Press, Inc. Vassiliou A. and Wickerhauser M.V Comparison of wavelet image coding schemes for seismic data compression. In: Wavelet Applications in Signal and Image Processing V. (eds Akram Aldroubi, Andrew F. Laine and Michael A. Unser). Proceedings of SPIE 3169, 118± 126. Villasenor J.P., Ergas R.A. and Donoho P.L Seismic data compression using highdimensional wavelet transforms. Proceedings of the Data Compression Conference, Snowbird, UT, USA, pp. 396±405. IEEE Computer Society Press. Wang B. and Pann K Kirchhoff migration of seismic data compressed by matching pursuit decomposition. 66th SEG meeting, Denver, USA, Expanded Abstracts, 1642±1645. Wu R.S. and Wang Y Seismic data compression using adapted local cosine transform and its effects on imaging. 61st EAGE conference, Helsinki, Finland, Extended Abstracts, P102.